Real analyticity for random dynamics of transcendental functions

Abstract

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh [On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748] leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron–Frobenius operator are assumed to converge. We also provide Bowen’s formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application establishes real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.